I have the following set:
$G=\{ z \in \mathbb{C} : \Im{(z)}>0, \Re{(z)}<0 \}$
$f(z) = z^2$
I need to draw $ f(G) $ but I don't get a good answer using $ z=x+iy $ and trying to understand the complex plane with information on the cartesian plane.
How do I go about it?
I understand that G is the upper left quarter of the complex plane and I know the answer is the lower half plane but I need to see the steps.
after that I need to write $ f(G) $. Would like to see that too.
Follow up question:
This time the function is $f(z) = log(z)$ and the set is:
$G=\{ z \in \mathbb{C} : |z|>0, -\pi < arg(z) < \pi \}$
I don't have an intuition of what the log function does to the given set so I'd appreciate a calculation.
Best Answer
We may rewrite
$$G=\{z\in\Bbb{C},|z|\gt 0,\,{\pi\over 2}\lt\arg{z}\lt\pi\}$$
Squaring means squaring the module and doubling the argument
So
$$f(G)=\{z\in\Bbb{C},|z|\gt 0,\,\pi\lt\arg{z}\lt 2\pi\}$$
And so $f(G)$ is the lower half plane
For the follow up consider the following
$$\log{r\cdot e^{i\theta}}=\log{r}+i \theta$$
This means that the image by logarithm of $H$ the right half plane is the horizontal strip $\{z\in\Bbb{C},-\pi\lt\operatorname{Im}{z}\lt\pi\}$